Modifying one end of a spiral/curve
You may do it using a curve as a Taper Object's target to change the thickness along the spiral's axis. You can then change its width using the curve's specials menu.


After creating a curve add a Bezier Circle and set it as a target in the Bevel Object window (Curve Data header).


Add a Bezier Curve and edit it in Edit Mode so it's straight and placed vertical.


Select the spiral in Object Mode and set the newly added Bezier Curve as a Taper Object target.


Modify the newly added curve to change the thickness along the spiral's axis. Experiment with it for different results.



Finally select the spiral in Object Mode, press W-->Width Size and drag the cursor away to change its width.